Math.Manifold.Embedding.Simple.Class

Some manifolds are “naturally” embedded within some bigger space. For instance, the topological spheres are readily identified with the geometric unit spheres in real vector spaces.

An embedding is a pretty strong relationship, but often all that's needed is being able to map single points from the manifold to the enclosing space. This module offers a class which does just that.

Documentation

class NaturallyEmbedded m v where

Methods

embed :: m -> v

coEmbed :: v -> m

Instances

NaturallyEmbedded ℝ ℝ

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝ -> ℝ

coEmbed :: ℝ -> ℝ

(VectorSpace y, VectorSpace z) => NaturallyEmbedded x ((x, y), z)

Defined in Data.Manifold.Types.Primitive

Methods

embed :: x -> ((x, y), z)

coEmbed :: ((x, y), z) -> x

VectorSpace y => NaturallyEmbedded x (x, y)

Defined in Data.Manifold.Types.Primitive

Methods

embed :: x -> (x, y)

coEmbed :: (x, y) -> x

AdditiveGroup f => NaturallyEmbedded x (FibreBundle x f)

Defined in Data.Manifold.FibreBundle

Methods

embed :: x -> FibreBundle x f

coEmbed :: FibreBundle x f -> x

(RealFloat s, VectorSpace s, s' ~ s) => NaturallyEmbedded (S⁰_ s) s'

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S⁰_ s -> s'

coEmbed :: s' -> S⁰_ s

(RealFloat s, VectorSpace s, s' ~ s) => NaturallyEmbedded (D¹_ s) s'

Defined in Data.Manifold.Types.Primitive

Methods

embed :: D¹_ s -> s'

coEmbed :: s' -> D¹_ s

(Num s, s ~ s') => NaturallyEmbedded (V2 s) (V2 s')

Defined in Data.Manifold.Types.Primitive

Methods

embed :: V2 s -> V2 s'

coEmbed :: V2 s' -> V2 s

(Num s, s ~ s') => NaturallyEmbedded (V3 s) (V3 s')

Defined in Data.Manifold.Types.Primitive

Methods

embed :: V3 s -> V3 s'

coEmbed :: V3 s' -> V3 s

(Num s, s ~ s') => NaturallyEmbedded (V4 s) (V4 s')

Defined in Data.Manifold.Types.Primitive

Methods

embed :: V4 s -> V4 s'

coEmbed :: V4 s' -> V4 s

(Num s, s ~ s') => NaturallyEmbedded (ZeroDim s) (ZeroDim s')

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ZeroDim s -> ZeroDim s'

coEmbed :: ZeroDim s' -> ZeroDim s

(RealFloat s, s' ~ s) => NaturallyEmbedded (S¹_ s) (V2 s')

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S¹_ s -> V2 s'

coEmbed :: V2 s' -> S¹_ s

(RealFloat s, s' ~ s) => NaturallyEmbedded (S²_ s) (V3 s')

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S²_ s -> V3 s'

coEmbed :: V3 s' -> S²_ s

(RealFloat s, s' ~ s) => NaturallyEmbedded (ℝP²_ s) (V3 s')

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝP²_ s -> V3 s'

coEmbed :: V3 s' -> ℝP²_ s

(Real s, NaturallyEmbedded x p, s ~ Scalar (Needle x)) => NaturallyEmbedded (Cℝay x) (p, s)

Defined in Data.Manifold.Types.Primitive

Methods

embed :: Cℝay x -> (p, s)

coEmbed :: (p, s) -> Cℝay x

(AdditiveGroup y, AdditiveGroup g) => NaturallyEmbedded (FibreBundle x f) (FibreBundle (x, y) (f, g))

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle x f -> FibreBundle (x, y) (f, g)

coEmbed :: FibreBundle (x, y) (f, g) -> FibreBundle x f

(NaturallyEmbedded m v, VectorSpace f) => NaturallyEmbedded (FibreBundle m (ZeroDim s)) (FibreBundle v f)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle m (ZeroDim s) -> FibreBundle v f

coEmbed :: FibreBundle v f -> FibreBundle m (ZeroDim s)

(NaturallyEmbedded v w, s' ~ s) => NaturallyEmbedded (FibreBundle (V2 s) v) (FibreBundle (V2 s') w)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle (V2 s) v -> FibreBundle (V2 s') w

coEmbed :: FibreBundle (V2 s') w -> FibreBundle (V2 s) v

(NaturallyEmbedded v w, s' ~ s) => NaturallyEmbedded (FibreBundle (V3 s) v) (FibreBundle (V3 s') w)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle (V3 s) v -> FibreBundle (V3 s') w

coEmbed :: FibreBundle (V3 s') w -> FibreBundle (V3 s) v

(NaturallyEmbedded v w, s' ~ s) => NaturallyEmbedded (FibreBundle (V4 s) v) (FibreBundle (V4 s') w)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle (V4 s) v -> FibreBundle (V4 s') w

coEmbed :: FibreBundle (V4 s') w -> FibreBundle (V4 s) v

(RealFloat s, InnerSpace s, s ~ s', s ~ s'', s ~ s''') => NaturallyEmbedded (FibreBundle (S¹_ s) s') (FibreBundle (V2 s'') (V2 s'''))

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle (S¹_ s) s' -> FibreBundle (V2 s'') (V2 s''')

coEmbed :: FibreBundle (V2 s'') (V2 s''') -> FibreBundle (S¹_ s) s'

(RealFloat' s, InnerSpace s, s ~ s', s ~ s'', s ~ s''') => NaturallyEmbedded (FibreBundle (S²_ s) (V2 s')) (FibreBundle (V3 s'') (V3 s'''))

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle (S²_ s) (V2 s') -> FibreBundle (V3 s'') (V3 s''')

coEmbed :: FibreBundle (V3 s'') (V3 s''') -> FibreBundle (S²_ s) (V2 s')

NaturallyEmbedded v w => NaturallyEmbedded (FibreBundle ℝ v) (FibreBundle ℝ w)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle ℝ v -> FibreBundle ℝ w

coEmbed :: FibreBundle ℝ w -> FibreBundle ℝ v